Motivation Decision Problems Functional Representation H´ ajek’s Basic Logic: Decision and Representation Simone Bova bova@unisi.it Department of Mathematics and Computer Science University of Siena (Italy) December 19, 2007 Logic Seminar University of Barcelona (Spain) Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Decision Problems Functional Representation Outline Motivation 1 Vague Notions Basic Logic Decision Problems 2 Derivability and Validity Complexity and Algorithms Functional Representation 3 Free Algebras and Normal Forms Open Problems Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation Outline Motivation 1 Vague Notions Basic Logic Decision Problems 2 Derivability and Validity Complexity and Algorithms Functional Representation 3 Free Algebras and Normal Forms Open Problems Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation Sorite’s Paradox X i ⇋ “a collection of i grains is a heap”, N ⇋ 1000000 Tentative axiomatization of the notion of heap ( i = 0 , . . . , N − 1): (H1) X N (H2) ¬ X 0 (H3. i ) X i + 1 → X i Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation Sorite’s Paradox X i ⇋ “a collection of i grains is a heap”, N ⇋ 1000000 Tentative axiomatization of the notion of heap ( i = 0 , . . . , N − 1): (H1) X N (H2) ¬ X 0 (H3. i ) X i + 1 → X i The theory is inconsistent: 1 X N 2 X N − 1 . . . . . . N + 1 X 0 N + 2 ¬ X 0 Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation Bivalence versus Vagueness We can either reject vagueness . . . 0 = X 0 = · · · = X 500000 < X 500001 = ... = X N = 1 “500001 grains form a heap, whether 500000 do not” Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation Bivalence versus Vagueness We can either reject vagueness . . . 0 = X 0 = · · · = X 500000 < X 500001 = ... = X N = 1 “500001 grains form a heap, whether 500000 do not” . . . or abjure bivalence: 0 = X 0 < X 1 < ... < X N − 1 < X N = 1 “ i grains form a heap” is less true than “ j grains form a heap”, if i < j Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation H´ ajek’s Paradigm | Fuzzy Logic Fuzzy logics are propositional logics over ⊤ , ⊥ , ⊙ , → st: variables X , Y , . . . are interpreted over [ 0 , 1 ] ; ⊤ and ⊥ are interpreted over 1 and 0; ⊙ and → are interpreted over binary functions on [ 0 , 1 ] ; ¬ X ⇋ X → ⊥ . Fuzzy conjunction and implication must maintain: the behavior of Boolean counterparts over { 0 , 1 } 2 ; intuitive properties of Boolean counterparts over [ 0 , 1 ] 2 ; the validity of fuzzy modus ponens . Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation H´ ajek’s Paradigm | Boolean Logic Intuitive properties of Boolean conjunction and implication: 1 1 y y Boolean conjunction is 0 0 1 commutative, associative, weakly increasing in both arguments, and has 1 as unit. 0 0 0 x x 1 1 1 1 y y Boolean implication, x implies y , 0 0 1 is 1 iff x ≤ y , weakly decreasing in x , weakly increasing in y . 0 0 0 x x 1 1 Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation H´ ajek’s Paradigm | t -Norms and Residua Definition (Continuous t -Norm, Residuum) A continuous t-norm ⊙ ∗ is a continuous binary function on [ 0 , 1 ] that is associative, commutative, monotone ( x ≤ y implies x ⊙ ∗ z ≤ y ⊙ ∗ z ) and has 1 as unit ( x ⊙ ∗ 1 = x ). Given a continuous t -norm ⊙ ∗ , its residuum is the binary function → ∗ on [ 0 , 1 ] defined by x → ∗ y = max { z : x ⊙ ∗ z ≤ y } . t -norms and their residua provide suitable interpretations for fuzzy conjunction and implication. Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation H´ ajek’s Paradigm | G¨ odel Logic ⊙ G and → G over [ 0 , 1 ] 2 : 1 1 x ⊙ G y = min ( x , y ) 0 0 0 y x x 0 1 � 1 if x ≤ y 1 1 x → G y = y otherwise 0 0 0 y x x 0 1 Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation H´ ajek’s Paradigm | Łukasiewicz Logic ⊙ L and → L over [ 0 , 1 ] 2 : 1 1 x ⊙ L y = max ( 0 , x + y − 1 ) 0 0 0 y x x 0 1 1 1 x → L y = min ( 1 , − x + y + 1 ) 0 0 0 y x x 0 1 Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation Basic Logic | Logical Calculus ⊢ BL φ iff φ is derivable in the following Hilbert calculus: (A1) ( φ → χ ) → (( χ → ψ ) → ( φ → ψ )) (A2) ( φ ⊙ χ ) → φ (A3) ( φ ⊙ χ ) → ( χ ⊙ φ ) (A4) ( φ ⊙ ( φ → χ )) → ( χ ⊙ ( χ → φ )) (A5) (( φ ⊙ χ ) → ψ ) ↔ ( φ → ( χ → ψ )) (A6) (( φ → χ ) → ψ ) → ((( χ → φ ) → ψ ) → ψ ) (A7) ⊥ → φ (R1) φ, φ → χ ⊢ χ Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Vague Notions Decision Problems Basic Logic Functional Representation Basic Logic | Semantic Completeness BL is the logic of all continuous t -norms and their residua [Cignoli et al., 2000]: ( i ) ⊢ BL χ iff, for every t -norm ⊙ ∗ and every assignment v , χ evaluates to 1 with respect to ⊙ ∗ and v . ( ii ) φ 1 , . . . , φ n ⊢ BL χ iff, for every t -norm ⊙ ∗ and every assignment v , if φ 1 , . . . , φ n evaluate to 1 with respect to ⊙ ∗ and v , then χ evaluates to 1 with respect to ⊙ ∗ and v . Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Derivability and Validity Decision Problems Complexity and Algorithms Functional Representation Outline Motivation 1 Vague Notions Basic Logic Decision Problems 2 Derivability and Validity Complexity and Algorithms Functional Representation 3 Free Algebras and Normal Forms Open Problems Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Derivability and Validity Decision Problems Complexity and Algorithms Functional Representation Derivability and Validity Let φ 1 , . . . , φ n , χ be formulas over X 1 , . . . , X n . BL-CONS n = {� ( { φ 1 , . . . , φ m } , { χ } ) � : φ 1 , . . . , φ m ⊢ BL χ } BL-TAUT n = {� χ � : ( ∅ , { χ } ) ∈ BL-CONS n } ⊆ BL-CONS n Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Derivability and Validity Decision Problems Complexity and Algorithms Functional Representation Derivability and Validity Let φ 1 , . . . , φ n , χ be formulas over X 1 , . . . , X n . BL-CONS n = {� ( { φ 1 , . . . , φ m } , { χ } ) � : φ 1 , . . . , φ m ⊢ BL χ } BL-TAUT n = {� χ � : ( ∅ , { χ } ) ∈ BL-CONS n } ⊆ BL-CONS n There are infinitely many t -norms and infinitely many assignments of n propositional variables over [ 0 , 1 ] . Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Derivability and Validity Decision Problems Complexity and Algorithms Functional Representation Derivability and Validity Let φ 1 , . . . , φ n , χ be formulas over X 1 , . . . , X n . BL-CONS n = {� ( { φ 1 , . . . , φ m } , { χ } ) � : φ 1 , . . . , φ m ⊢ BL χ } BL-TAUT n = {� χ � : ( ∅ , { χ } ) ∈ BL-CONS n } ⊆ BL-CONS n There are infinitely many t -norms and infinitely many assignments of n propositional variables over [ 0 , 1 ] . Question: Is BL-TAUT n decidable? And BL-CONS n ? Simone Bova H´ ajek’s Basic Logic: Decision and Representation
Motivation Derivability and Validity Decision Problems Complexity and Algorithms Functional Representation Generic t -Norms | 2-Variate Fragment 3 [ 0 , 1 ] MV = ([ 0 , 3 ] , ⊙ 2 , → 2 , ⊥ ) : 8 max ( x 1 + x 2 − 1 , 0 ) if 0 ≤ x 1 , x 2 < 1 > > > max ( x 1 + x 2 − 2 , 1 ) if 1 ≤ x 1 , x 2 < 2 3 < x 1 ⊙ 2 x 2 = 2 3 max ( x 1 + x 2 − 3 , 2 ) if 2 ≤ x 1 , x 2 ≤ 3 1 > > 2 0 > min ( x 1 , x 2 ) if ⌊ x 1 ⌋ � = ⌊ x 2 ⌋ : 0 0 x_2 1 1 1 2 2 x_1 x_1 0 3 8 3 if x 1 ≤ x 2 > > > x 2 − x 1 + 1 if 0 ≤ x 1 , x 2 < 1 > > 3 < x 1 → 2 x 2 = x 2 − x 1 + 2 if 1 ≤ x 1 , x 2 < 2 2 3 1 > x 2 − x 1 + 3 if 2 ≤ x 1 , x 2 ≤ 3 2 > 0 > > 0 0 > x_2 : x 2 if ⌊ x 2 ⌋ < ⌊ x 1 ⌋ 1 1 1 2 2 x_1 x_1 0 3 Simone Bova H´ ajek’s Basic Logic: Decision and Representation
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